Problem: What is the slope of the line tangent to $f(x) = 2x^{2}+4x-6$ at $x = -3$ ?
Answer: The slope of the tangent line is $ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(2(x+\Delta x)^{2}+4(x+\Delta x)-6) - (2x^{2}+4x-6)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(2(x^{2}+2x \Delta x+\Delta x^{2})+4(x+\Delta x)-6) - (2x^{2}+4x-6)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{2x^{2}+4(x \Delta x)+2\Delta x^{2}+4x+4(\Delta x)-6-2x^{2}-4x+6}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{4(x \Delta x)+2\Delta x^{2}+4(\Delta x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} 4x+2(\Delta x)+4$ $ = 4x+4$ $ = (4)(-3)+4$ $ = -8$